The log-plurisubharmonicity of fiberwise $\xi-$Bergman kernels for variant functionals

Abstract

In this paper, we establish the log-plurisubharmonicity of fiberwise $\xi$-Bergman kernels for a family of variant functionals, thereby addressing a question posed by Bo Berndtsson to the authors. As an application, we prove that for a plurisubharmonic function $\phi$ and a locally finitely generated ideal sheaf $\mathscr{I}$ on the polydisc $\Delta^{n+m}=\Delta^n\times\Delta^m$, the set of points $w\in\Delta^m$ for which $\mathcal{I}(\phi|_{H_w})_{(o,w)}\subseteq (\mathscr{I}|_{H_w})_{(o,w)}$ is a pluripolar set, where $H_w$ represents the fiber over $w$, and $\mathcal{I}(\phi|_{H_w})$ denotes the multiplier ideal sheaf of $\phi|_{H_w}$ on $H_w$.